Motional Emf

Consider a simple circuit in which a conducting rod of length $l$ slides along a U-shaped conducting frame in the presence of a uniform magnetic field. This circuit is illustrated in Figure 2.31. Suppose, for the sake of simplicity, that the magnetic field is directed perpendicular to the plane of the circuit. To be more exact, the magnetic field is directed into the page in the figure. Suppose, further, that the rod moves to the right at the constant speed $v$.

Figure 2.31: Motional emf.

The magnetic flux passing through the circuit is simply the product of the perpendicular magnetic field-strength, $B$, and the area of the circuit, $l\,x$, where $x$ determines the position of the sliding rod. Thus,

$${\mit\Phi}_B = B\,l\,x.$$ (2.395)

Now, the rod moves a distance $dx=v\,dt$ in a time interval $dt$, so in the same time interval the magnetic flux passing through the circuit increases by

$$d{\mit\Phi}_B = B\,l\,dx = B\,l\,v\,dt.$$ (2.396)

It follows, from Faraday's law [see Equation (2.284)], that the magnitude of the emf $V$ generated around the circuit is given by

$$V= \frac{d{\mit\Phi}_B}{d t} = B\,l\,v.$$ (2.397)

Thus, the emf generated in the circuit by the moving rod is simply the product of the magnetic field-strength, the length of the rod, and the velocity of the rod. If the magnetic field is not perpendicular to the circuit, but instead subtends an angle $\theta$ with respect to the normal direction to the plane of the circuit, then it is easily demonstrated that the so-called motional emf generated in the circuit by the moving rod is

$$V = B_\perp\,l\,v,$$ (2.398)

where $B_\perp = B\,\cos\theta$ is the component of the magnetic field that is perpendicular to the plane of the circuit.

Because the magnetic flux linking the circuit increases in time, by Lenz's law, the emf acts in the negative direction (i.e., in the opposite sense to the fingers of a right-hand, if the thumb points along the direction of the magnetic field). The emf, $V$, therefore, acts in a counter-clockwise direction in the figure. If $R$ is the total resistance of the circuit then this emf drives an counter-clockwise electric current of magnitude $I=V/R$ around the circuit. Of course, this current generates a magnetic field that acts to reduce the increase in the magnetic flux passing through the circuit.

But, where does the motional emf come from? Let us again remind ourselves what an we mean by an emf. When we say that an emf $V$ acts around the circuit in the counter-clockwise direction, what we really mean is that a charge $q$ that circulates once around the circuit in a counter-clockwise direction acquires the energy $q\,V$. The only manner in which the charge can acquire this energy is if something does work on it as it circulates. Let us assume that the charge circulates very slowly. The magnetic field exerts a negligibly small force on the charge when it is traversing the non-moving part of the circuit (because the charge is moving very slowly). However, when the charge is traversing the moving rod it experiences an upward (in the figure) magnetic force of magnitude $f=q\,v\,B$ (assuming that $q>0$). (See Section 2.2.4.) The net work done on the charge by this force as it traverses the rod is

$$W' = q\,v\,B\,l = q\,V,$$ (2.399)

because $V = B\,l\,v$. Thus, it would appear that the motional emf generated around the circuit can be accounted for in terms of the magnetic force exerted on charges traversing the moving rod.

However, there is something seriously wrong with the previous explanation. We seem to be saying that the charge acquires the energy $q\,V$ from the magnetic field as it moves around the circuit once in a counter-clockwise direction. But, this is impossible, because a magnetic field cannot do work on an electric charge. (See Section 2.2.4.)

Let us look at the problem from the point of view of a charge $q$ traversing the moving rod. In the frame of reference of the rod, the charge only moves very slowly, so the magnetic force acting on it is negligible. In fact, only an electric field can exert a significant force on a slowly moving charge. In order to account for the motional emf generated around the circuit, we need the charge to experience an upward force of magnitude $q\,v\,B$. The only way in which this is possible is if the charge sees an upward pointing electric field of magnitude

$$E = v\,B.$$ (2.400)

In other words, although there is no electric field in the laboratory frame, there is an electric field in the frame of reference of the moving rod, and it is this field that does the necessary amount of work on charges moving around the circuit in order to account for the existence of the motional emf, $V=B\,l\,v.$

More generally, if a conductor moves in the laboratory frame with velocity ${\bf v}$ in the presence of a magnetic field ${\bf B}$ then a charge $q$ inside the conductor experiences a magnetic force ${\bf f} = q\, {\bf v}\times{\bf B}$. In the frame of the conductor, in which the charge is essentially stationary, the same force takes the form of an electric force ${\bf f} = q\,{\bf E}$, where ${\bf E}$ is the electric field in the frame of reference of the conductor. Thus, if a conductor moves with velocity ${\bf v}$ through a magnetic field ${\bf B}$ then the electric field ${\bf E}$ that appears in the rest frame of the conductor is given by

$${\bf E} = {\bf v} \times {\bf B}.$$ (2.401)

(See Section 3.4.1.) This electric field is the ultimate origin of the motional emfs that are generated whenever circuits move with respect to magnetic fields.